# Weak homotopy equivalences

## Idea

A weak homotopy equivalence is a map between topological spaces or simplicial sets or similar which induces isomorphisms on all homotopy groups. In homological algebra this is called a quasi-isomorphism.

The localization or simplicial localization of the categories Top and sSet at the weak homotopy equivalences used as weak equivalences yields the standard homotopy category Ho(Top) and Ho(sSet) or the (∞,1)-category of ∞-groupoids/homotopy types, respectively.

Weak homotopy equivalences are named after homotopy equivalences. They an be identified with homotopy equivalences after one allows to replace the domains by a resolution. The corresponding notions in homological algebra are quasi-isomorphisms and chain homotopy-equivalences.

From another perspective, the notion of weak homotopy equivalence is ‘observational’, in that a map is a weak homotopy equivalence if when we look at it through the observations that we can make of it using homotopy groups or even the fundamental infinity-groupoid, it looks like an equivalence. In contrast, homotopy equivalence is more ‘constructive’; in that $f$ is a homotopy equivalence if there exists an inverse for it (up to homotopy, of course). Note that both of these notions are weaker than mere isomorphism of topological spaces (homeomorphism) and so can be considered examples of weak equivalences.

There are actually two related concepts here: whether two spaces are weakly homotopy equivalent and whether a map between spaces is a weak homotopy equivalence. The former is usually defined in terms of the latter.

## Definition (for topological spaces and simplicial sets)

###### Definition

For $X, Y \in$ Top or $\in$ sSet two topological spaces or simplicial sets, a continuous function or simplicial map $f : X \to Y$ between them is called a weak homotopy equivalence if

1. $f$ induces an isomorphism of connected components

$\Pi_0(f) \colon \Pi_0(X) \stackrel{\simeq}{\to} \Pi_0(Y)$

in Set;

2. for all all points $x \in X$ and for all $(1 \leq n) \in \mathbb{N}$ $f$ induces an isomorphism on homotopy groups

$\pi_n(f,x) \colon \pi_n(X,x) \stackrel{\simeq}{\to} \pi_n(Y,f(x))$

in Grp.

###### Remark

If $X$ and $Y$ are path-connected, then (1) is trivial, and it suffices to require (2) for a single (arbitrary) $x$, but in general one must require it for at least one $x$ in each path connected component.

###### Definition

The homotopy category of Top with respect to weak homotopy equivalences is Ho(Top)${}_{whe}$.

###### Remark

Accordingly, weak homotopy equivalences are the weak equivalences in the classical “Quillen” model structure on topological spaces and the model structure on simplicial sets, and also in the mixed model structure.

## Properties

### Equivalent characterizations

###### Proposition

A continuous map $f : X \to Y$ is a weak homotopy equivalence precisely if for all $n \in \mathbb{N}$ and for all commuting diagrams of continuous maps of the form

$\array{ S^{n-1} &\to& X \\ \downarrow && \downarrow^{\mathrlap{f}} \\ D^n &\to& Y } \,,$

where the left morphism is the inclusion of the $(n-1)$-sphere as the boundary of the $n$-ball, there exists a continuous map $\sigma : D^n \to X$ that makes the resulting upper triangle commute and such that the lower triangle commutes up to a homotopy

$\array{ S^{n-1} &&\to&& X \\ \\ \downarrow && \nearrow & \swArrow& \downarrow^{\mathrlap{f}} \\\\ D^n &&\to&& Y }$

which is constant along $S^{n-1} \hookrightarrow D^n$.

In this form the statement and its proof appears in (Jardine) (where it is also generalized to weak equivalences in a model structure on simplicial presheaves). See also around (Lurie, prop. 6.5.2.1). The relevant arguments are spelled out in (May, section 9.6). A variant is called the HELP lemma in (Vogt).

### Relation to homotopy equivalences

###### Proposition

Every homotopy equivalence is a weak homotopy equivalence.

###### Proof

It requires a little bit of thought to prove this, because $f$ and its homotopy inverse $g$ need not preserve any chosen basepoint. But for any $x\in X$ and any $n\ge 1$, we have a diagram

$\array{\pi_n(X,x) & & \to & & \pi_n(X,g(f(x)))\\ & \searrow && \nearrow && \searrow\\ && \pi_n(Y,f(x)) && \to && \pi_n(Y,f(g(f(x))))}$

in which the two horizontal maps are isomorphisms because $g f$ and $f g$ are homotopic to identities. Hence, by the two-out-of-six property for isomorphisms, the diagonal maps are also all isomorphisms.

###### Proposition

Conversely, any weak homotopy equivalence between m-cofibrant spaces (spaces that are homotopy equivalent to CW complexes) is a homotopy equivalence.

### Relation to homotopy types

We discuss the equivalence relation generated by weak homotopy equivalence, called (weak) homotopy type. For the “abelianized” analog of this situation see at quasi-isomorphism the section Relation to homology type.

###### Proposition

The existence of a weak homotopy equivalence from $X$ to $Y$ is a reflexive and transitive relation on Top, but it is not a symmetric relation.

###### Proof

Reflexivity and transitivity are trivially checked. A counterexample to symmetry is example 1 below.

But we can consider the genuine equivalence relation generated by weak homotopy equivalence:

###### Definition

We say two spaces $X$ and $Y$ habe the same (weak) homotopy type if they are equivalent under the equivalence relation generated by weak homotopy equivalence.

###### Remark

Equivalently this means that $X$ and $Y$ have the same (weak) homotopy type if there exists a zigzag of weak homotopy equivalences

$X \leftarrow \to\leftarrow \dots \to Y \,.$

This in turn is equivalent to saying that $X$ and $Y$ become isomorphic in the homotopy category Ho(Top)/Ho(sSet) with the weak homotopy equivalences inverted.

###### Remark

Two spaces $X$ and $Y$ may have isomorphic homotopy groups without being weak homotopy equivalence: for this all the isomorphisms must be induced by an actual map $f : X \to Y$, as in the above definition.

However, if, roughly, one remembers, how all the homotopy groups act on each other, then this is enough information to exhibit the full homotopy type. This collection of data is called the Postnikov tower decomposition of a homotopy type.

## For other kinds of spaces

A map of simplicial sets is called a weak homotopy equivalence equivalently if its geometric realization is a weak homotopy equivalence of topological spaces, as above. (Since the geometric realization of any simpicial set is a CW complex, in this case its geometric realization is actually a homotopy equivalence.)

Likewise, a functor between small categories is sometimes said to be a weak homotopy equivalence if its nerve is a weak homotopy equivalence of simplicial sets, hence of topological spaces after geometric realization of categories. These are the weak equivalences in the Thomason model structure on categories (not the canonical model structure). The statement of Quillen's theorem A and Quillen's theorem B in in this contex.

Similarly, one can define weak homotopy equivalences between any sort of object that has a geometric realization, such as a cubical set, a globular set, an n-category, an n-fold category, and so on.

Note that in some of these cases, such as as simplicial sets, symmetric sets, and probably cubical sets, there is also a notion of “homotopy equivalence” from which this notion needs to be distinguished. A simplicial homotopy equivalence, for instance, is a simplicial map $f:X\to Y$ with an inverse $g:Y\to X$ and simplicial homotopies $X\times \Delta^1 \to X$ and $Y\times \Delta^1 \to Y$ relating $f g$ and $g f$ to identities.

A different direction of generalization is the notion of a homotopy equivalence of toposes.

## Examples

### Of non-reversible weak homotopy equivalences

We discuss examples of weak homotopy equivalences that have no weak homotopy equivalence going the other way, according to prop. 4 above.

###### Example

Let $S^1 \in$ Top denote the ordinary circle and $\mathbb{S}$ the pseudocircle.

There is a continuous function $S^1 \to \mathbb{S}$ which is a weak homotopy equivalence, hence in particular $\pi_1(\mathbb{S}) \simeq \mathbb{Z}$. But every continuous map the other way round has to induce the trivial map on $\pi_1$.

This is the simplest in a class of counter-examples discussed in (McCord).

## References

A general account is for instance in section 9.6 of

The characterization of weak homotopy equivalences by lifts up to homotopy seems is in

• Rick Jardine, Simplicial Presheaves, Journal of Pure and Applied Algebra 47, 1987, no.1, 35-87.